HINT: <no title>
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You need to plot the graphs of the two equations on
the same Cartesian plane.
STEP:
Plot the graph of
y=x2+6x+5
on a Cartesian plane
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We must plot the graphs of the two equations on the same
Cartesian plane. We will then need to find the points where the two graphs
intersect. We will begin by plotting graph of the quadratic equation:
y=x2+6x+5.
To plot this graph, we need to find the coordinates of the
points where the graph will cross the x- and
y-axes. These points are the x- and
y-intercepts, respectively. We will then use these points
to plot the graph.
The y-intercept is the quantity in the
equation which is not multiplied by x.
For this equation, the y-intercept is
5.
The coordinates of this point are
(0;5).
The x-intercepts are the roots of the
equation
x2+6x+5=0.
We will calculate the roots of the equation
through factorisation. In this case, we will use the grouping method.
For this method, we need two numbers that add to give
6.
This is the coefficient of the middle term.
The product of the two numbers must be
5. This is the product of
the coefficient of the first term 1 and
the value of the last term 5. The two numbers are
1 and
5. We will re-write the middle term,
forming two terms, using these two numbers.
We could use the quadratic formula as well. In this case,
it is not necessary to do so because we can factorise the equation.
00=x2+6x+5=x2+x+5x+5
We now group the first two terms together and group the last terms
together and factorise. After that, we will factor the highest common factor.
00=x(x+1)+5(x+1)=(x+5)(x+1)
If we equate each binomial to zero we get the roots of the
quadratic equation. The roots are
x=−1 and
x=−5. The coordinates of these
roots are (−1;0) and
(−5;0).
Using these x- and
y-intercepts we will draw the graph:
STEP:
Plot the graph of y=2x+2 on the
same Cartesian plane
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As we did for the quadratic equation, we will first
find the y-intercept. After that we will
calculate root for the linear equation. We remember
that the
y-intercept is the quantity in the
equation which is not multiplied by x.
The
y-intercept is
2. Its coordinates are:
(0;2).
We will work out the root of the equation
0=2x+2.
02xx=2x+2=−2=−1=0
The root is: x=−1. It's
coordinates are: (−1;0).
We will now plot the equation y=2x+2 on the same Cartesian plane we have used
for the graph of the quadratic equation.
STEP: Read off the coordinates of the points of intersection
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As we can see from the second graph, there are two points of
intersection. We have marked each of these points with a black
dot. The coordinates of these points are:
(−1;0) and
(−3;−4).
We have used a graphical method to determine the solutions of
the simultaneous equations:
y=x2+6x+5 and
y=2x+2. The solutions are
the points where their graphs intersect.
It is possible to solve the simultaneous equations using
either the substitution or elimination methods.
The points of intersection are
(−1;0) and
(−3;−4).
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